Theorem prnz 4718

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4703	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4276	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2109   ≠ wne 2944  Vcvv 3430  ∅c0 4261  {cpr 4568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-sn 4567  df-pr 4569

This theorem is referenced by:  opnz  5390  propssopi  5424  fiint  9052  wilthlem2  26199  upgrbi  27444  wlkvtxiedg  27972  shincli  29703  chincli  29801  spr0nelg  44880  sprvalpwn0  44887